Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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Chapter 4 Nonlinear Modeling of Complex Waveforms in the f − x Domain 4.1 Linear Prediction in the f − x Domain F - X deconvolution is a popular noise attenuation tool first introduced by Canales (1984). Seismic traces are represented in the time-space domain. When one trans- forms each trace into a Fourier domain, the complex waveforms are represented in the frequency-space domain (f − x). Linear events are predicted for each frequency in the spatial direction of a given frequency. Therefore, if a signal can be predicted, the difference between the observed and predicted signals can be considered an estimation of the noise in the data. Linear prediction filters that map to monochromatic complex sinusoids in the f− x domain can accurately predict linear events in the t − x domain. A superposition of complex harmonics immersed in white noise can be predicted using an ARMA (autoregressive-moving average) model as suggested by Sacchi and Kuehl (2001). 47
4.1. LINEAR PREDICTION IN THE F − X DOMAIN 48 An ARMA model can be approximated by a long autoregressive model (AR), which turns out to be a representation of the linear prediction problem. In summary, linear events in t−x space transform into complex sinusoids in the f −x domain and linear prediction filtering can properly model the spatial variability of the waveforms at any given monochromatic temporal frequency f. The seismic signal is considered to be an AR model; let us assume a single waveform in time domain. In addition, let’s assume that the signal has linear moveout s(x, t) = a(t − xθ) (4.1) where x is offset of the trace, t is the time and θ is the slowness of the event. In the frequency domain this signal becomes S(x, f) = A(f)e −i2πfθx (4.2) where A(f) denotes the source spectrum and f is the temporal frequency for x. By discretizing x = (j − 1)δx, f = fl Sjl = Ale −i2πflθ(j−1)δx (4.3) I can develop this model as a function of wave number by fixing kl = 2πflθ equation (4.3) becomes Sjl = Ale −ikl(j−1)δx (4.4) I can define the problem by predicting data along the each trace to fix temporal
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Name of Author: Soner Bekleric Univ
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University of Alberta Faculty of Gr
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Abstract Linear filter theory has p
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Contents 1 Introduction 1 1.1 Appli
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List of Tables 3.1 Filter length an
Page 11 and 12: 4.4 (a) Prediction of Figure 4.2(b)
Page 13 and 14: List of symbols Symbol Name or desc
Page 15 and 16: List of abbreviations Abbreviation
Page 17 and 18: In applied seismology predictive de
Page 19 and 20: 1.1. APPLICATIONS 4 applied to vide
Page 21 and 22: 1.3. THESIS OUTLINE 6 • Chapter 4
Page 23 and 24: 2.2. LINEAR PROCESS 8 Finally, I pr
Page 25 and 26: 2.3. LINEAR PREDICTION 10 E(z) 1 X(
Page 27 and 28: 2.3. LINEAR PREDICTION 12 In my the
Page 29 and 30: 2.3. LINEAR PREDICTION 14 which can
Page 31 and 32: 2.3. LINEAR PREDICTION 16 ∂ρ fb
Page 33 and 34: 2.3. LINEAR PREDICTION 18 I have an
Page 35 and 36: 2.4. 1-D SYNTHETIC AND REAL DATA EX
Page 39 and 40: 2.6. SUMMARY 24 Relative PSD Relati
Page 41 and 42: Chapter 3 Nonlinear Prediction 3.1
Page 43 and 44: 3.1. NONLINEAR PROCESSES VIA THE VO
Page 45 and 46: 3.1. NONLINEAR PROCESSES VIA THE VO
Page 47 and 48: 3.2. NONLINEAR MODELING OF TIME SER
Page 61: 3.4. SUMMARY 46 3.4 Summary In this
Page 65 and 66: 4.2. ANALYSIS OF OPTIMUM FILTER LEN
Page 67 and 68: 4.2. ANALYSIS OF OPTIMUM FILTER LEN
Page 69 and 70: 4.3. NONLINEAR PREDICTION OF COMPLE
Page 73 and 74: RMSE 2 4.3. NONLINEAR PREDICTION OF
Page 77 and 78: 4.4. NOISE REMOVAL AND VOLTERRA SER
Page 79 and 80: 4.5. SYNTHETIC AND REAL DATA EXAMPL
Page 89 and 90: 4.6. SUMMARY 74 4.6 Summary In this
Page 91 and 92: 5.1. INTRODUCTION 76 ples remains a
Page 93 and 94: 5.3. SYNTHETIC DATA EXAMPLES 78 Not
Page 95 and 96: 5.3. SYNTHETIC DATA EXAMPLES 80 Tim
Page 97 and 98: 5.3. SYNTHETIC DATA EXAMPLES 82 Tim
Page 99 and 100: 5.4. REAL DATA EXAMPLES 84 Time [s]
Page 105 and 106: 5.5. SUMMARY 90 5.5 Summary In this
Page 107 and 108: series in this thesis has been trun
Page 109 and 110: 6.1. FUTURE DIRECTIONS 94 Noise red
Page 111 and 112: BIBLIOGRAPHY 96 Canales, L. L. (198
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BIBLIOGRAPHY 98 Marple, S. L. (1980
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BIBLIOGRAPHY 100 Trad, D. (2003). I
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Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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